Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance

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Manifest Covariance

by

Paul Henry Williams

Dissertation presented for the degree of Doctor of Philosophy

in Physics in the Faculty of Natural Science at Stellenbosch

University

Supervisors: Prof. F.G. Scholtz Dr. J.N. Kriel March 2020

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Declaration

By submitting this dissertation electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the sole author thereof (save to the extent explicitly otherwise stated), that reproduction and publication thereof by Stellenbosch University will not infringe any third party rights and that I have not previously in its entirety or in part submitted it for obtaining any qualification.

Date: . . . .

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Abstract

Lorentz

Symmetry in Non-Commutative Field Theories:

Commutative/Non-Commutative

Dualities and Manifest

Covariance

Paul Henry Williams

Department of Physics, University of Stellenbosch,

Private Bag X1, Matieland 7602, South Africa.

Dissertation: PhD Physics March 2020

We cover two approaches to building Lorentz invariant non-commutative field theories.

First we construct dualities between commutative and non-commutative field t heories. T his c onstruction e xploits a g eneralization o f t he e xact renor-malization group equation (ERG). We review ERG dualities for the two di-mensional quantum mechanical Landau problem. We also review the idea of non-canonical field t heories. From t his we b uild a n E RG d uality f or t he free non-canonical complex scalar field t heory. T his a pproach a llows u s t o track the Lorentz symmetry, we show this explicitly for the free theory. Finally we construct dualities for the φ4 _{interacting theory.}

Second we build a manifestly Lorentz invariant 2 + 1 dimensional field theory living in SU(1, 1) fuzzy space-time. Here the commutation relations themselves respect the Lorentz symmetry. We start by reviewing the non-relativistic construction of SU(2) fuzzy space and quantum mechanics. We briefly d iscuss a p otential fi eld th eory ex tension of th is. We th en be gin our SU(1, 1) construction by reviewing the SU(1, 1) group and make the connection to space-time. We then build a quantum theory living on this space-time and introduce dynamics with the Klein-Gordon equation. From this we can move on to a scalar field theory in terms of functions on the group m anifold. Finally we make the connection to commutative theories by introducing the symbols of operators. From this we are able to compute different correlators and compare with commutative theories.

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Uittreksel

Lorentz

Simmetrie in Nie-kommutatiewe Veldeteorieë:

Kommutatiewe/Nie-kommutatiewe

Dualiteite en

Eksplisiete

Kovariansie

(“Lorentz Symmetry in Non-Commutative Field Theories: Commutative/Non-Commutative Dualities and Manifest Covariance”)

Paul Henry Williams

Departement van Fisikia, Universiteit van Stellenbosch,

Privaatsak X1, Matieland 7602, Suid Afrika.

Proefskrif: PhD Fisikia Maart 2020

Ons beskryf twee benaderings tot die konstruksie van Lorentz invariante nie-kommutatiewe veldeteorieë.

Eerstens beskryf ons dualiteite tussen kommutatiewe en nie-kommutatiewe veldeteorieë. Hierdie konstruksie berus op ’n veralgemening van die Eksakte Renormerings Groep (ERG). Ons hersien dualiteite vir die twee-dimensionele kwantum meganiese Landau probleem. Dit word opgevolg deur ’n hersiening van nie-kanoniese veldeteorieë. Met dit as basis, konstrueer ons ’n ERG du-aliteit vir die vrye nie-kanoniese komplekse skalare veld. Hierdie benadering stel ons in staat om die Lorentz simmetrie na te volg, en dit word eksplisiet vir die vrye teorie gedemonstreer. Ten slotte, konstrueer ons dualiteite vir die φ4 wisselwerkende teorie.

Tweedens, konstrueer ons ’n eksplisiete Lorentz invariante 2 + 1 dimensio-nele veldeteorie in SU(1,1) wasige ruimte-tyd. In hierdie geval respekteer die kommutasieverbande self die Lorentz simmetrie. Ons begin met ’n hersiening van die nie-relativistiese konstruksie van wasige ruimte en die formulering van kwantum meganika op die ruimtes. Ons bespreek dan kortliks ’n veldteore-tiese veralgemening hiervan. Ons begin dan met die SU(1,1) konstruksie deur die hersiening van die SU(1,1) groep en die konneksie met ruimte-tyd. Ons konstrueer dan ’n kwantum veldeteorie op die ruimte-tyd en voer dinamika in deur die Klein-Gordon vergelyking. Met dit as basis, kan ons dan ’n vrye skalare teorie opbou in terme van funksies op die groep manifold. Ten slotte

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maak ons die verbintenis met kommutatiewe toerië deur die invoering van die simbole van operatore. Die gebruik hiervan stel ons in staat om verskillende korrelators te bereken en met kommutatiewe teorieë te vergelyk.

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Acknowledgements

I would like to express my sincere gratitude to:

My supervisors Prof. F.G. Scholtz and Dr J.N. Kriel, for all their guidance and support, without them this PhD would not have been possible. I will always value your dedication to my PhD project. My parents and family for all the love and support during my PhD and also my entire academic career. Thank you for getting me though this long and winding road, I am so grateful. NITheP and the Physics department of Stellenbosch for financial support during my PhD. My friends for the camaraderie that got us through some tough times, to those who are still busy studying good luck, you will get there in the end. Future students who read this, remember you’re in this together look after each other no one else understands your troubles better than your fellow students. The Stellenbosch Biokinetics Centre who gave me the opportunity and encouragement to improve my physical well being during my PhD. I am quite happy to not only have gotten some mental gains during my PhD but also physical ones as well.

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Chapter 1 Introduction

The structure of time at short length scales and the emergence of space-time as we perceive it at long length scales are probably the most challenging problems facing modern physics [1]. In order to combine gravity and quantum mechanics into a unified theory, we will need a more sophisticated understand-ing of physics at very short length scales. The compellunderstand-ing arguments of Do-plicher et al [2] highlighted the need for a revised notion of space-time at short length scales and gave strong arguments in favour of a non-commutative geom-etry. It is theorized that any attempt to localize events at a scale shorter than the Planck length will lead to gravitational instability. Naturally we should seek to build this maximal localization, or rather minimum length scale, into our theories. One realization of this is non-commutative space-time. Non-commutative space-time has received considerable attention in the past few decades, however, it is not a new idea. It was originally proposed by Snyder [3] in an attempt to avoid the ultra-violet divergences of field theories. The discovery of renormalization pushed these ideas to the background until more recently when they resurfaced in the search for a consistent theory of quantum gravity. Non-commutative coordinates also occur quite naturally in certain string theories [4], generally perceived to be the best candidate for a theory of quantum gravity.

This sparked renewed interest in non-commutative space-time and the for-mulation of quantum mechanics [5], and quantum field theories [6], on such spaces. Even though there have been many investigations into the possible physical implications of non-commutativity in quantum mechanics, quantum many-body systems [7; 8; 9; 10; 11], quantum electrodynamics [12; 13; 14], the standard model [15] and cosmology [16; 17], a systematic formulation of non-commutative quantum mechanics was lacking, until [5] where an operator val-ued formulation for two dimensional quantum mechanics was developed. Here the standard quantum mechanical formalism and interpretation was shown to carry over. The only modification required is that more general Hilbert spaces are needed to represent the states of the system, and to find the unitary rep-resentations of the non-commutative Heisenberg algebra. Here the simplest

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form of non-commutativity

[ˆx, ˆy] = iθ, (1.0.1)

for the space-time commutation relations was adopted, where θ is a parameter related to the minimum length scale. This is what we, from here on, refer to as the constant commutation relations, where the coordinates commutate to a multiple of the identity, when discussing types of non-commutative theories. These commutation relations break rotational and Lorentz symmetry. For field theories living in this space-time a twisted implementation of the Lorentz symmetry was suggested in [12].

The twisted implementation of the Lorentz symmetry [18] is still contro-versial and gave rise to several outstanding issues. The first difficulty is to carry out the standard Noether analysis and identify conserved charges for the twisted Lorentz symmetry. The correct quantization of these theories is also debated in the literature. Should one use the standard quantization procedure or also deform the canonical commutation relations? On the level of the path integral this means we need to alter the measure. In fact UV/IR mixing may be related to a quantum anomaly of the Lorentz group [18].

In this thesis we investigate two attempts at building a systematic and consistent framework for non-commutative field theories. There have been many studies of non-commutative field theories [6; 19; 20], however, we seek a framework where properties like symmetries, in particular Lorentz symmetry, unitary and renormalizabilty are manifest and interactions can be included naturally. Often these elements are incorporated in an ad-hoc way. The two methods we investigate use two very different approaches to ensure that the Lorentz symmetry is intact. Both of them, however, provide interesting results. The first approach we follow is based on the theories defined in [21; 22], often referred to as canonical quantum field theories. In these papers non-commutativity is implemented on the level of modified canonical commutation relations of the fields and momenta, rather than the space-time coordinates as is conventionally done [6]. This is close in spirit to non-commutative quan-tum mechanics as formulated in [5], where the degrees of freedom are the non-commutative coordinate and momenta. In non-canonical quantum field theory the Lorentz symmetry also gets violated, but a partial mapping to a commutative theory can restore the Lorentz invariance. However, this intro-duces non-local interactions for interacting theories and it is an open question whether these theories are unitary and renormalizable [23]. In [24] the exact renormalization group (ERG) was used to construct dualities between com-mutative and non-comcom-mutative quantum systems in two dimensions for the constant commutations relations. This construction was carried out for dif-ferent quantum systems, in particular the Landau problem. We seek to gen-eralize this to non-canonical quantum field theories. We note that although the resulting non-commutative quantum field theories may be non-local and

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CHAPTER 1. INTRODUCTION 3

not obviously renormalizable, unitary or Lorentz invariant, the duality must ensure that these properties are still present, albeit in a non-manifest way. A possible benefit of these dualities may be the weakening of interactions.

This is the motivation for part 1, which aims to demonstrate and construct commutative/non-commutative dualities for field theories using the ERG pro-gram. The focus here is on dualities between commutative and non-canonical quantum field theories as described in [21; 22; 23]. These dualities are straight-forward generalizations of the ones constructed in [24].

Part 1 is therefore organized as follows: in chapter 2 we review non-commutative quantum mechanics with constant commutation relations and non-canonical field theories. Chapter 3 reviews and expands on the duality construction of [24] for the Landau problem. Finally in chapter 4 we discuss the duality construction for non-canonical theories, we investigate the fate of the symmetries and construct the duality for an interacting theory.

The second approach we follow is based on the fuzzy space constructions as explored in [25]. In three dimensional fuzzy space the coordinates obey the su(2) algebra. Here the commutation relations clearly preserve rotational symmetry. This is in a similar spirit to non-commutative quantum mechanics as formulated in [5], but this time in the sense that we can build a similar op-erator valued formulation. In these papers a unitary representation of SU(2) is found on the non-commutative Hilbert space, instead of the Heisenberg al-gebra as was done in 2 dimensions. These commutation relations were studied for various systems including the free particle [26], spherical well [27] and hy-drogen atom [28]. However, these theories currently do not have a relativistic extension. In these theories time is commutative, which is incompatible with a relativistic theory where time and space should be on an equal footing. There have been attempts to introduce non-commutative time with constant commu-tation relations [29], but we seek a theory where the commucommu-tation relations are manifestly Lorentz invariant i.e. a fuzzy space-time. This is what we proceed to construct in part 2.

Part 2 attempts to build a non-commutative field theory living on a Lorentz invariant non-commutative fuzzy space-time. In 2 + 1 dimensions the most natural set of Lorentz invariant commutation relations would be su(1, 1), so we will build our space-time using these commutation relations. We also look at a SU(2) field theory as a prototype for our construction. We note that a slightly different construction of Snyder-space has already been explored in [30]. We begin by building a relativistic non-commutative quantum mechanical theory, as was done for the non-relativistic case in [25], and then use this to build a field theory.

Part 2 is organized as follows: in chapter 5 we review three dimensional fuzzy space quantum mechanics and discuss possible field theory constructions. In chapter 6 we review the group SU(1, 1) and discuss the connection to fuzzy space-time. In chapter 7 we discuss how we can build our relativistic non-commutative quantum mechanics and move on to a field theory. Finally, in

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chapter 8 we describe ways in which we can connect our non-commutative theory to measurable quantities and commutative theories.

Before we begin the thesis proper, we should spend some time reviewing the standard "textbook" descriptions of quantum physics, as these are the theories we wish to adapt in an attempt to create a modified description of physics at short length scales. We must make sure that our modifications disappear in the limits where the standard theories match measurements well i.e. long length scales. For the entire thesis we work in natural units with

~ = c = 1. (1.0.2)

1.1 Commutative Quantum Mechanics

We begin with non-relativistic quantum mechanics. In quantum mechanics, unlike classical physics, we cannot find the description of a particle’s location as a function of time, rather if we know the quantum state of a system, we can extract the information to be compared with measurements. Textbooks which cover the topics in this section in detail are [31; 32].

Mathematically we represent the state φ of our system as a vector in a Hilbert space H. The Hilbert space is a complete inner product vector space. We have the usual mathematical structure on the Hilbert space, namely an inner product

(·, ·) : H ⊗ H → C. (1.1.1)

A linear functional is defined as a linear map from the Hilbert space to the complex numbers

F : H → C. (1.1.2)

The space of linear functionals denoted by H∗ _{is also a Hilbert space, called}

the dual space of H. We know from Riesz’s theorem that we can write any linear functional as an inner product, so for every vector in the dual space there exists a linear functional

Fφ(ψ) = (φ, ψ), (1.1.3)

and vice versa. We can then use the Dirac bra-ket notion where we denote vectors in H by |ψi kets, and vectors in the dual space by bras Fφ ≡ hφ|. In

this notation the inner product is written as hφ|ψi ≡ (φ, ψ).

Now that we have the states of our system we need to introduce the ob-servables i.e. the quantities we are able to measure. From the correspondence principle we know that associated with every observable is a linear hermitian operator. These operators act on the states in our Hilbert space

ˆ

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We can define the adjoint of an operator,

(φ, ˆAψ) = ( ˆA†φ, ψ), (1.1.5)

then an operator is hermitian when ˆA = ˆA† and the domain of A is the same as the domain of A†_{. We can also define outer products |ψi hφ| which}

are operators.The eigenstates of hermitian operators, ˆA |ai = a |ai, form an orthonormal basis for the Hilbert space and all their eigenvalues are real a ∈ R. Via the spectral theorem we can write, if the spectrum is discrete

ˆ

A =X

a

a |ai ha| (1.1.6)

or if the spectrum is continuous ˆ A =

Z

da a |ai ha| . (1.1.7)

To extract the physical information from these states and observables, we begin with a more general construction: mixed states. If we say the system is in particular mixed state we mean it has probability pi of being in a state

|ψii. We can then define the density matrix as

ˆ

ρ =X

i

pi|ψii hψi| . (1.1.8)

Now we have introduced not only quantum but also statistical uncertainty into the system. If the system is known to be in a state ψnthen the density matrix

is simply

ˆ

ρ = |ψni hψn| . (1.1.9)

This is known as a pure state. When we make a measurement of an observable A, the possible values we can obtain are the eigenvalues a. The probability P (a) of obtaining the value a when measuring the observable A for a system with a density matrix ˆρ is

P (a) = ha| ˆρ |ai . (1.1.10)

In this way we introduce the probabilistic interpretation of quantum mechan-ics, we extract the probabilities of measuring values of observables from the state the system is in. We see even for pure states we have the quantum uncer-tainty. We note this uncertainty is not a statement about our ignorance like statistical uncertainty, but rather is a fundamental property of nature. This is known as a strong measurement, and after these measurements the wave-function collapses into the state |ai associated with the value measured. To make a sensible probabilistic interpretation requires our states to be normal-ized. Therefore we require the states in the Hilbert space to have finite norm i.e. hφ|φi < ∞.

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Two typical observables are position ˆx and momentum ˆp. These operators satisfy the Heisenberg algebra:

[ˆxi, ˆpj] = iδij. (1.1.11)

All other commutators are zero, namely [ˆxi_{, ˆ}_xj_{] = [ˆ}_pi_{, ˆ}_pj_{] = 0}_{. This means that}

we cannot find simultaneous eigenstates for these operators, since they do not commute, and we cannot simultaneously measure these observables. From this commutator we know that we have an uncertainty principle for ˆx and ˆp

∆x∆p ≥ 1

2. (1.1.12)

Where the ∆A = qh ˆA2_{i − h ˆ}_Ai2 is the standard deviation of an observable.

When quantizing a classical system a key step is to find a unitary representa-tion of the Heisenberg algebra. Since ˆ~x is a hermitian operator we can use its eigenstates |~xi as a basis for our Hilbert space. Note x is a continuous label. We can find a unitary representation of the Heisenberg algebra, commonly referred to as the Schrödinger representation

ˆ ~ x |~xi = ~x |~xi , ˆ ~ p |~xi = −i ∂ ∂~x|~xi , (1.1.13) then [xi_{, −i∂}

xj] = iδij. In the position basis the inner product is R d3x φ∗(~x)ψ(~x)

and the Hilbert space is now L2_{, which are the square integrable functions, as}

required for a sensible probabilistic interpretation. The Stone-von Neumann theorem ensures that this is a unique representation up to unitary transfor-mations.

Another important set of operators are the orbital angular momentum operators, which are the generators of rotations. In three dimensions the rotation transformations are elements of SO(3). Then the angular momentum operators satisfy the su(2) algebra

[ ˆLi, ˆLj] = iijkLˆk. (1.1.14)

This implies the uncertainty

∆Li∆Lj ≥

1 2| hijk

ˆ

Lki |. (1.1.15)

There is a well-known homomorphism between SO(3) and SU(2). In fact SU(2) is a double cover of SO(3). This is how spin is introduced. Spin being an angular momentum observable with no classical analogue. Then we have operators for spin Si and total angular momentum Ji, which also satisfy the

su(2) algebra. The position representation of the orbital angular momentum is given by the cross product

ˆ ~ L |~xi = 1 i~x × ∂ ∂~x|~xi . (1.1.16)

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1.1.1 The Schrödinger Equation

We can introduce dynamics into our quantum theory. Non-relativistically the time-dependent Schrödinger equation governs how states change in time:

i∂

∂t|ψ, ti = ˆH |ψ, ti . (1.1.17)

ˆ

H is the Hamiltonian, it contains the information about the factors which influence our dynamics. For a particle interacting with a static electromagnetic field the Hamiltonian, from minimal substitution, is ˆH = ( ˆp−q ~~ _2mA(ˆ~x))2 + qV (ˆ~x). The time-dependent Schrödinger equation is then

i∂ ∂t|ψ, ti = (ˆ~p − q ~A(ˆ~x))2 2m + qV (ˆ~x) ! |ψ, ti . (1.1.18)

If we multiply from the left by h~x| in the position basis i∂ ∂tψ(~x, t) = (−i ~∂x− q ~A(~x))2 2m + qV (~x) ! ψ(~x, t). (1.1.19)

If the Hamiltonian is not time dependent we can write ψ(x, t) = e−i ˆHt_ψ(x),

then we choose ψ(x) to be an eigenstate of ˆH with eigenvalue E. This then produces the time-independent Schrödinger equation

Eψ(~x) = (−i ~∂x− q ~A(~x))

2

2m + qV (~x)

!

ψ(~x). (1.1.20)

This is an eigenvalue equation for E, which determines the allowed energies of our system.

ψ(~x) is a function ψ(~x) : R3 _{→ C. We call R}3 _{the configuration space,}

which is also a Hilbert space, the space that the physical system occupies. We note that free particle solutions ei~k·~x are not actually in the Hilbert space.

They are limit points of this space and thus extra care must be taken when making a probabilistic interpretation.

1.1.2 The Path Integral

The propagator is defined as

K(x0, t0, xf, tf) = hxf, tf|x0, t0i . (1.1.21)

We can calculate this by the usual time slicing method that yields in the continuous limit the path integral in phase space

K(x0, t0, xf, tf) = Z (xf,tf) (x0,t0) [dx][dp] ei Rtf t0 dt p ˙x−H(p,x)_. (1.1.22)

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If we perform the [dp] integral we obtain K(x0, t0, xf, tf) = Z (xf,tf) (x0,t0) [dx] eiS, (1.1.23) where S = Z tf t0 dt ˙x2 2m − V (x) . (1.1.24)

This is an integral over all the configurations the function x(t) can take i.e. over all the possible paths a particle can take from x0 = x(t0) to xf = x(tf).

We note that this is the classical action. In fact, the path integral is one way to quantize classical systems by starting with a classical action.

We can calculate normalized time-ordered expectation values (correlators) using the path integral

hx0, t0| T [ˆx(tn) . . . ˆx(t1)] |xf, tfi hx0, t0|xf, tfi = R(xf,tf) (x0,t0) [dx][dp] x(tn) . . . x(t1)e iRtf t0 dt p ˙x−H(p,x) K(x0, t0, x, t) . (1.1.25)

If we perform the appropriate analytic continuation of time and then take the limit where tf → ∞e−iφ and t0 → −∞e−iφ this projects out the ground state,

then we can use this path integral to calculate vacuum to vacuum expectation values. One way to do this is the Wick rotation here φ = π

2.

We can define the normalized generating functional by adding a source to the action Z[J ] = R(xf,tf) (x0,t0) [dx] e iS+Rtf t0 dt J (t)x(t) R(xf,tf) (x0,t0) [dx] e iS , (1.1.26)

often written in terms of the unnormalized generating functional Z[J] = W [J ] W [0].

Then we can compute any correlator by taking functional derivatives with respect to the normalized generating functional

hx0, t0| T [ˆx(tn) . . . ˆx(t1)] |xf, tfi hx0, t0|xf, tfi = δ J (tn) . . . δ J (t1) Z[J ] J =0 . (1.1.27)

From now on we are only interested in vacuum to vacuum correlators.

As an example let us consider the Landau problem. In 2 dimensions we introduce complex coordinates z(t) = x(t) + iy(t). For a particle of charge −e < 0 moving in a magnetic field B > 0 pointing in the positive z-direction, the action in the symmetric gauge is given by

S = Z ∞ −∞ dt m ˙ z∗₊ ieB 2mz ∗ ˙z −ieB 2mz −e 2_B2 4m z ∗ z . (1.1.28)

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Electromagnetic interactions were added by minimal substitution. The sim-plest way to quantize this action is to follow [32]. For this it is convenient to return to the phase-space functional integral for which the action in the commutative case is given by, in the symmetric gauge,

S = Z ∞ −∞ dt π∗ ˙z − ieB 2mz + ˙ z∗₊ ieB 2mz ∗ π −1 mπ ∗ π − e 2_B2 4m z ∗ z . (1.1.29)

This gives the commutation relations

ˆz, ˆπ† = ˆz†, ˆπ = i, (1.1.30) with all other commutators vanishing. Using these commutation relations the Hamiltonian is found to be ˆ H = 1 mπˆ † ˆ π + e 2_B2 4m zˆ † ˆ z +ieB 2m πˆ † ˆ z − ˆz†π −ˆ eB 2m, (1.1.31)

where care with ordering is taken so that we get the correct form for the Zeeman term. We can factorize ˆH as

ˆ H = 1 m ˆ π†− ieB 2 zˆ † ˆ π + ieB 2 zˆ − eB 2m. (1.1.32)

By introducing the boson creation and annihilation operators ˆ_{b =} _√1 eB ˆ π†− ieB 2 zˆ † , ˆb† = √1 eB ˆ π + ieB 2 zˆ (1.1.33) we can write ˆ H = ωc ˆ_b†_ˆ b + 1 2 (1.1.34) where ωc = eB_m is the cyclotron frequency. As usual there is an infinite

de-generacy in each Landau level. This is due to another set of creation and annihilation operators, constructed in terms of the guiding centre coordinates, which commute with ˆH.

1.2 Commutative Field Theory

Now we can move on to a relativistic version of quantum mechanics. Textbooks which cover the topics in this section in detail are [33; 34].

We can write down a relativistic version of the Schrödinger equation, mo-tivated by the relativistic dispersion relation

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When electromagnetic interactions are included the corresponding equation, obtained from minimal substitution, is given by

((∂ − qA(xν))µ(∂ − qA(xν))µ− m2)ψ(t, ~x) = 0. (1.2.2)

This equation, which describes spin-0 particles, is called the Klein-Gordon equation, the Dirac equation for spin-1

2 particles can be obtained by linearizing

the dispersion relation. In this thesis we only deal with spin-0 theories. Due to problematic features, such as negative norms for the solutions of the Klein-Gordon equation and negative energies in both the Klein-Gordon and Dirac theories, the single particle probabilistic interpretation of ψ fails. A proper interpretation, as fields, can be obtained when we consider many-particle systems i.e. quantum field theories.

We start with a classical field theory in 3 + 1 dimensions. We have a classical Lagrangian, which now depends on the classical field φ(~x, t) depending on coordinate labels. We can introduce the momentum conjugate to this field, usually π = ∂L

∂ ˙φ. From this we can find the classical Hamiltonian. For the

complex scalar field this is H =

Z

d3x π∗(~x, t)π(~x, t)+( ~∇φ∗(~x, t))·( ~∇φ(~x, t))+m2φ∗(~x, t)φ(~x, t). (1.2.3) If we write the path integral action

S = Z

d4x φ∗(~x, t)−∂_t2+ ~∇2− m2φ(~x, t), (1.2.4) we can quantize using the standard method of Dirac brackets or the method presented in [35]. After quantizing the field operators satisfy the Heisenberg algebra (at equal times)

[ ˆφ(~x, t), ˆπ(~y, t)] = iδ(~x − ~y), (1.2.5) [ ˆφ(~x, t), ˆφ(~y, t)] = [ˆπ(~x, t), ˆπ(~y, t)] = 0. (1.2.6) Moving to the quantum field theory, we consider our observables ˆφ and ˆπ to be operators which act on a Hilbert space. Now these fields are not our physical states but rather our observables and coordinates are now relegated to labels. The Hilbert space the ˆφ and ˆπ will act on is a Fock space with multi-particle states. We can write our field in terms of creation and annihilation operators

ˆ φ(~x, t) = Z d3k hei~k·~xaˆ~_k(t) + e−i~k·~xˆb_~†_k(t) i . (1.2.7)

Here the ˆa~_k and ˆb~_k create and destroy particle and anti-particle states with

momentum ~k. We can diagonalize the Hamiltonian as ˆ H = Z d3k hωkˆa † ~ k(t)ˆa~k(t) + ωk ˆ_b† ~_k(t)ˆb~k(t) + ωk i , (1.2.8)

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where ωk =

√

k2_{+ m}2. We see that the Hamiltonian counts all the momentum

contributions to the energy for the different particles and anti-particles. We also see the divergent vacuum contribution. Every point in space acts like the ground state of a harmonic oscillator, even though the space contains no particles. Since space is continuous even in a finite volume the vacuum contribution is divergent. This divergence is something we hope to remove by introducing a minimum length scale, like in our non-commutative theories.

We can find a "position" basis similar to the quantum mechanical case in terms of eigenstates of the fields |φ(x, t)i, the eigenvalues are possible field configurations φ(x, t),

hφ|F i = F [φ]. (1.2.9)

Now F [φ] is a functional of the field configuration φ. The representation of the Heisenberg algebra on these functionals is given uniquely by

hφ| ˆφ |F i = φF [φ], hφ| ˆπ |F i = −i δ

δφF [φ]. (1.2.10)

We can think of the quantum mechanical case as field theory in 0+1 dimensions with coordinates xi(t)being fields discretely labeled by i.

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Part I

Non-commutative/Commutative

Dualities

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Chapter 2 Non-Commutative and

Non-Canonical Theories

In part 1 our goal is to develop dualities between different non-canonical field theories, as defined in [21], and standard field theories. By dualities we mean that the generating functionals of the two theories are the same and hence also all the correlators. It could also allow us to track symmetries, and other properties such as unitarity and renormalizability. It could give insight into how to correctly implement the symmetries in non-canonical theories, about which there is still debate [12]. Finally, it could provide a systematic way to introduce interactions in non-canonical theories, the correct way to do this is not obvious [21; 22]. Another potential advantage as mentioned in the intro-duction, is that these dualities could make it easier to calculate quantities, by calculating them for the dual systems first. One way this simplifies calculations is by weakening interactions in the dual system.

In this chapter we review non-commutative quantum mechanics, specifi-cally focusing on the two-dimensional case. This is what we generalize when we study non-canonical theories. After this we review non-canonical field the-ories, as our aim is to build dualities for these theories.

2.1 Non-Commutative Quantum Mechanics

We believe there must exist a minimum length scale and that we should build this into our theories. In non-commutative theories our coordinates are oper-ators, then their commutation relations introduce, from the uncertainty prin-ciple, a minimum length scale. In two dimensions, the simplest choice for the commutation relations are the constant commutation relations:

[ˆxi, ˆxj] = iijθ. (2.1.1)

This leads to the uncertainty

∆xi∆xj ≥ |

ij_θ|

2 , (2.1.2)

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where θ, which has the dimension of length squared, is related to the minimum length scale. These commutation relations where studied in [5], where an operator-based formulation for quantum mechanics was developed. It was found that mathematically it is structured exactly in the same way as standard quantum mechanics. It is then possible to find a unitary representation of the non-commutative Heisenberg algebra. The only modification required is that more general Hilbert spaces are needed to represent the states of the system. Since the coordinates do not commute we cannot simultaneously measure both coordinates. The way to deal with this is to introduce weak, rather then strong, measurements when position is measured.

The non-commutative coordinate algebra (2.1.1) acts on a Hilbert space Hc, which we call the configuration space or classical Hilbert space. We use

the angular ket for states in this space: |ni ∈ Hc. We can realize the

con-stant commutation relations by introducing a pair of creation and annihilation operators, ˆ_{b =} _√1 2θ(ˆx + iˆy), ˆ b†= √1 2θ(ˆx − iˆy), (2.1.3) such that [ˆb, ˆb†

] = ˆIc. The configuration space is then simply the boson Fock

space Hc = span{|ni = 1 √ n(ˆb † )n|0i}n=∞_n=0 . (2.1.4)

Now our quantum states (wave-functions) are operators which are elements of the algebra generated by the coordinate operators i.e. ψ(ˆx1_{, ˆ}_x2₎_{. These}

states live in a Hilbert space called the quantum Hilbert space. Note that this is different from the configuration space. The most natural choice for the quantum Hilbert space is the space of Hilbert-Schmidt operators.

Hq = {ψ(ˆx1, ˆx2) : Trc(ψ†(ˆx1, ˆx2)ψ(ˆx1, ˆx2)) < ∞}. (2.1.5)

We use a subscript q for the quantum Hilbert space to distinguish it from the configuration space Hc. The finite trace condition is a generalization of

square integrability. However, we can still consider scattering theory [36], in the same way as commutatively, by considering operators outside this Hilbert space (free particle solutions), which are limit points of Hq. We use the rounded

ket to represent states | ˆψ) ∈ Hq and reserve the symbol † specifically for the

operator adjoint on Hc. The natural inner product on this space is

( ˆφ| ˆψ) = Trc(φ†(ˆx1, ˆx2)ψ(ˆx1, ˆx2)). (2.1.6)

As in standard quantum mechanics we have linear functionals and a dual space which are defined using the trace inner product.

Now we need to introduce observables i.e. operators which act on states in the quantum Hilbert space. The non-commutative Heisenberg algebra acting

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CHAPTER 2. NON-COMMUTATIVE AND NON-CANONICAL THEORIES 15

on the quantum Hilbert space, obey the commutation relations [ ˆXi, ˆXj] = iijθ,

[ ˆXi, ˆPi] = iδij,

[ ˆPi, ˆPi] = 0. (2.1.7)

We use capital letters for these operators ˆXi_{, ˆ}_Pi _{to distinguish them from the}

coordinate operators acting on configuration space. These are often called super operators since they operate on operators. We use the symbol ‡ for the adjoint on Hq defined with respect to the trace inner product

( ˆφ| ˆO ˆψ) = ( ˆO‡φ| ˆˆψ). (2.1.8) We can find a unitary representation of the non-commutative Heisenberg algebra in the following way [5]

ˆ Xi| ˆψ) = |ˆxiψ),ˆ ˆ Pi| ˆψ) = 1 θ ij_|[ˆ_xj_{, ˆ}_ψ]). _(2.1.9)

This is similar to the position representation of commutative quantum me-chanics. The non-commutative time independent Schrödinger equation is

(P − q ~~ˆ A(X))~ˆ 2 2m + qV ( ˆ ~ X) ! | ˆψ) = E| ˆψ), (2.1.10) where electromagnetic interactions have been added by minimal substitution as in the commutative case.

Instead of the operator-based formulation above, the constant commutation relations can also be realized in terms of scalar functions. This is similar to the Wigner phase space construction of quantum mechanics using a star product [37] and quasiprobabilities. However, this is much less physically intuitive. A star product which implements the non-commutative coordinate algebra is the Moyal product given by

∗M = e i 2θ ij←_∂−i−→_∂j , (2.1.11) xi∗M xj− xj ∗M xi = iijθ. (2.1.12)

Then the scalar functions depend on the numbers xi_{, but all regular products}

of scalar functions now become composed with star products. However, these commuting coordinates cannot be interpreted as true physical coordinates; they have no clear physical meaning. This star product is also not unique, but there are infinitely many that would lead to the same commutation relations. Another star product commonly used is the Voros product

∗V = e i 2θij ←− ∂i−_∂→j₊i 2θ ←− ∂i−→_∂i . (2.1.13)

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In this formulation the time independent Schrödinger equation can be written in terms of functions as (ˆ~p − q ~A(ˆ~x))2 2m + qV (ˆ~x) ! ∗V /Mψ(x) = Eψ(x), (2.1.14)

Later it will become apparent why this is potentially the more natural choice. Mathematically these formulations are equivalent [38] , however, care must be taken with the physical interpretation of these star products and in particular the space of functions on which they are defined, which is not simply L2_{. These}

domains are even different for the Voros and Moyal products and great care must be taken when any comparison of theories built on these two products is made [38].

The Moyal product can also be thought of as a transformation that mixes position and momentum, since the star product shifts a coordinate by a deriva-tive. The coordinates in the scalar functions are then combinations of position and momentum and can therefore not be interpreted as coordinates in position space. The Bopp shift does exactly this on the operator level

ˆ xi = ˆx˜i+ i θ 2 ij_ˆ ˜ pj, (2.1.15)

this is one way to build our non-commutative algebra. Here the tilde operators are operators which obey the standard position and momentum commutation relations. It is then possible to build eigenstates for these tilde coordinates but they are not physical and care must be taken when using these states for calculations.

2.2 Position and the Path Integral

We would like to introduce a "position state", however, since we cannot find simultaneous eigenstates for ˆx and ˆy the best we can do is a minimum un-certainty state. It is possible to define a one mode Glauber coherent state in configuration space [39]

|zi = e−|z|22 ezˆb †

|0i . (2.2.1)

In the quantum Hilbert space we define a state, which is known to have the interpretation of a minimum uncertainty state [5]

|z) = |zi hz| , (2.2.2)

for θ > 0

z = √1

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is a dimensionless complex number. These states form an over-complete basis for the configuration space and it is possible to write the identity as

ˆ_I_c ₌ 1 π

Z

dzdz∗ |zi hz| . (2.2.4)

As these are minimum uncertainty states, the real and imaginary parts of z are the closest we can get to points in space obtained from a measurement. We can write the identity on the quantum Hilbert space as

ˆ_I q= 1 π Z dzdz∗ |z) ∗V (z|. (2.2.5)

The Voros product, which in terms of the complex coordinates is given by ∗V = e

←− ∂z

−→

∂z∗, is necessary for this to be the identity. Motivating why this is a

more natural choice than the Moyal product.

We can think of quantum mechanics as field theory in 0 + 1 dimensions, the coordinates are the fields which depend on time xi_(t)_{. A path integral}

approach to non-commutative quantum mechanics been developed in [40], we now mention the relevant results. Here we time slice using the |z) states, with our propagator being (zf, tf|z0, t0), then the calculation proceeds using the

completeness of these states in terms of the star product. This propagator is given by (zf, tf|z0, t0) = N e −∂_zf∂z∗ 0 Z (zf,tf) (z0,t0) [dz][dz∗]eiS. (2.2.6) As calculated in [40], for the free theory the action in the above propagator is given by S = Z tf t0 dt 1 2z˙ ∗ 1 2m + iθ 2∂t −1 ˙z. (2.2.7)

2.2.1 The Landau Problem

One of the simplest non-trivial dualities we can build is for the Landau problem [41]. This construction will also be useful for non-canonical field theories due to the similar form of the non-canonical and the non-commutative Landau problem actions. We now briefly review the non-commutative Landau problem. The action was derived in [40; 42], and reads

S = Z ∞ −∞ dt " 1 2 ˙ z∗₊ieB 2mz ∗ 1 2m + iθ 2∂t −1 ˙z − ieB 2mz −e 2_B2 4m z ∗ z . (2.2.8)

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In phase space the non-commutative action is given by S = Z ∞ −∞ dt π∗ ˙z − ieB 2mz + ˙ z∗₊ieB 2mz ∗ π −π∗ 1 m + iθ∂t π − e 2_B2 4m z ∗ z . (2.2.9)

The functional integral is now over π, π∗ _{and z, z}∗_.

The above action can be quantized, the simplest way to do so is to follow [35]. The commutation relations are found to be

ˆz, ˆπ† = ˆz†, ˆπ = i, [ˆz, ˆz†] = θ, (2.2.10) with all other commutators vanishing. This calculation is shown in more detail later in this chapter for the case of the non-canonical field theory. Now the momenta are still commuting, but the coordinates are non-commuting. The Hamiltonian is still given by (1.1.31) as in the commutative case. We can factorize in the same way as in (1.1.32). Now the creation and annihilation operators are ˆ_{b =} 1 q e| ˜B| ˆ π†− ie ˜B 2 zˆ † ! , ˆb†= _q1 e| ˜B| ˆ π + ie ˜B 2 zˆ ! , (2.2.11)

where the effective magnetic field, ˜ B = B 1 −eBθ 4 , (2.2.12)

is positive. For negative ˜B the definitions of ˆb and ˆb† are exchanged. The Hamiltonian becomes

HN C = ˜ωcˆb†ˆb +

˜ ωc

2 , (2.2.13)

where ˜ωc is the effective cyclotron frequency ˜ωc = e| ˜_mB|. There is again an

infinite degeneracy of each Landau level.

Now we have reviewed all we need to construct dualities for these non-commutative quantum mechanical theories. This is the subject of the next chapter. The rest of this chapter is devoted to reviewing non-canonical field theories, keeping in mind the ultimate goal of this part of the thesis is to construct dualities for non-canonical field theories.

2.3 Non-Canonical Field Theory

One naive way found in the literature to build non-commutative field theories, is to take standard Lagrangians and insert star products between the fields.

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This, however, produces several problems. Which star product to use is not obvious, and the space of fields is now no longer L2_{so one would have to modify}

the path integral measure [18], which is quite difficult. However, there are more systematic ways to build non-commutative field theories [42], but even then implementing the symmetries is not obvious [12]. We discuss in part 2 a way to build a theory in non-commutative space-time that, by construction, respects Lorentz symmetry, but for now we discuss a different kind of non-commutative theory.

In non-canonical field theories, space-time is not non-commutative, but rather we have commutative fields. This is more in the spirit of non-commutative quantum mechanics where the degrees of freedom are the coordi-nates. In field theories the fields are the degrees of freedom where as position is simply what we use to label the fields. We label the coordinates xi _{with i,}

while the fields φ(x) are labeled by x.

We change the canonical commutation relation of our observables, hence for a non-canonical field theory, we postulate, at equal times,

[ ˆΦa(x), ˆΦb(y)] = iθab(x − y), (2.3.1)

[ ˆΠa(x), ˆΠb(y)] = iδa,bδ(x − y). (2.3.2)

The a, b denote different species of fields and θab is some function of x −

y. These operators associated with the observable fields will be the super-operators, denoted by capitals. Following the quantum mechanical case they act on states in Hq. These states are operators which are elements of the

algebra generated by the non-commutative field configurations i.e. F [ˆφ] ≡ | ˆF ). The field configurations act on the configuration space |fi ∈ Hc. The

non-commutative field configurations satisfy modified commutation relations, at equal times,

[ ˆφa(x), ˆφb(y)] = iθab(x − y), (2.3.3)

and play the role of the non-commutative coordinates.

2.3.1 Complex Field Theory

Now we consider the complex scalar field theory i.e. a field theory with two different real fields. We consider non-commutativity of the form [21], at equal times,

[ ˆΦ(x), ˆΦ‡(y)] = θδ(x − y), (2.3.4)

the fields also obey the standard equal time commutation relations

[ ˆΦ(x), ˆΠ(y)] = iδ(x − y). (2.3.5)

Recalling that ‡ _{is the adjoint on the quantum Hilbert space. Again we}

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non-commutative quantum mechanical case, ˆ Φ(x, t)| ˆF ) = | ˆφ(x, t) ˆF ), (2.3.6) ˆ Π(x, t)| ˆF ) = 1 θ|[ ˆφ †_{(y, t), ˆ}_{F ]),} _(2.3.7)

the same for ‡ _fields.

All other commutation relations are standard. The free Hamiltonian is given by ˆ H = Z d3x ˆΠ‡(~x, t) ˆΠ(~x, t) + ( ~∇ ˆΦ‡(~x, t)) · ( ~∇ ˆΦ(~x, t)) + m2Φˆ‡(~x, t) ˆΦ(~x, t). (2.3.8) We can write it in Fourier modes, by introducing the Fourier transform

ˆ Φ(x) = Z d3k e−ikxΦk ˆ Π(x) = Z d3k e−ikxΠk (2.3.9) then we have ˆ H = Z d3k ˆΠ‡_kΠˆk+ ωk2Φˆ ‡ kΦˆk, (2.3.10)

these operators satisfy the following commutation relations

[ ˆΦk, ˆΠk0] = iδ(k + k0), (2.3.11) [ ˆΦ‡_k, ˆΠ‡_k0] = iδ(k + k 0 ), (2.3.12) [ ˆΦk, ˆΦ ‡ k0] = 2 gk ωk δ(k − k0), (2.3.13) where ωk= √ k2 _{+ m}2_, _g k = ωkθ 2 . (2.3.14)

We can diagonalize the Hamiltonian by introducing creation and annihilation operators acting on Hq, the details of this calculation are in [21],

ˆ H = Z d3k hωk p1 + g2 k+ gk ˆA ‡ kAˆk+ ωk p1 + g2 k− gk ˆB ‡ kBˆk +ωkp1 + g2k i . (2.3.15)

This has the very interesting property of matter anti-matter asymmetry, the physical consequences of this is discussed in [23].

The path integral action in position space is given by S = Z d4x φ∗(t, ~x) −∂2 t 1 + iθ∂t− i + ~∇2_{− m}2 φ(t, ~x). (2.3.16)

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This is derived in [21] using a method similar to the one discussed above, the difference is that the field theory equivalent of Bopp shift states (2.1.15) are used instead of coherent states. Note the similarity with (2.2.7).

We will need this action when we build our dual theory. As a consistency check, we quantize this theory to verify that the commutation relations (2.3.4) are indeed reproduced. Introducing auxiliary fields χ and χ∗ _{we can rewrite}

the action as S =

Z

d4x [χ∗(1 + iθ∂t− i) χ + χ∗∂tφ + χ∂tφ∗

+φ∗( ~∇2_{− m}2_)φi_. _(2.3.17)

The epsilon term ensures the convergence of the path integral, but hereafter we drop it. We follow the Faddeev-Jackiw method [35] to quantize the system. We denote

ξi = {φ, φ∗, χ, χ∗}, (2.3.18)

and write the Lagrangian as L = Z d3x 1 2ξ i_ω ij∂tξj − V (ξ) , (2.3.19) with ωij =     0 0 0 −1 0 0 −1 0 0 1 0 iθ 1 0 −iθ 0     . (2.3.20)

Following [35] the commutation relations (2.3.4) are indeed recovered from the entries of iωij _{with ω}ij _{the inverse of ω}

ij. Following quantization the

Hamiltonian has the form ˆ

H = V (ξ) = Z

d3x hχˆ‡χ − ˆˆ Φ‡( ~∇2_{− m}2_{) ˆ}_Φi_. _(2.3.21)

The auxiliary fields are constrained to the conjugate momenta, yielding the final Hamiltonian

ˆ H =

Z

d3x h ˆΠ‡Π + ~ˆ ∇ ˆΦ‡· ~∇ ˆΦ + m2Φˆ‡Φˆi. (2.3.22) This does indeed match the non-canonical Hamiltonian we started with.

2.4 Chapter Summary

We have reviewed the non-commutative theories we seek to develop dualities for. We do this because we believe that these non-commutative theories are

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a better description of nature at short length scales. However, we need the dualities to track symmetries and the renormalizabilty of the non-commutative theories. We covered the two dimensional quantum mechanical case including the path integral formulation. We also studied the very similar in spirit non-canonical field theories. We now can move on to constructing the dualities for our non-canonical field theories. First, however, we review the dualities for quantum mechanical theories, this will provide some additional insights in the next chapter.

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Chapter 3 The Exact Renormalization Group

(ERG) and Non-Commutative

Dualities

In this chapter we introduce and review dualities for the non-commutative quantum mechanical case of the Landau problem. The goal of part one is to generalize the ERG dualities to the non-canonical field theory case. With a duality we mean that the generating functionals of the two theories are the same and hence also all the correlators. Note, however, in the context of the ERG a weaker form of duality is normally implied in that the generating functionals and correlators are the same below a certain momentum scale.

In the case of a complex scalar field, the generating functional is given by Z[J ] = W [J ] W [0] (3.0.1) where W [J ] = Z [dφ∗][dφ]ei[S[φ∗,φ]+J∗φ+J φ∗]. (3.0.2) We discuss two ways to build these dualities. First we use a change of variables that mixes momentum and position similar to the Bopp shift. This is physically more intuitive and mathematically transparent, but it is difficult to apply this to systems with more general interactions. This is why we use the second method based on the exact renormalization group (ERG). In particular, the ERG will be useful when studying interacting systems, which we discuss in the next chapter.

3.1 Bopp Shift

We start with a generalized Bopp shift, and derive a slightly more general duality then just a commutative/non-commutative one. We introduce the

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change of variables

z = Z − iaπ, z∗ = Z∗+ iaπ∗_{, a ∈ R} (3.1.1)

into the non-commutative phase-space generating functional integral with ac-tion (2.2.9). The acac-tion then becomes

S = Z ∞ −∞ dt " π∗ Z −˙ ie ˜B 2 ˜mZ ! + Z˙∗₊ ie ˜B 2 ˜mZ ∗ ! π − π∗ 1 ˜ m + i˜θ∂t π −e 2_B_˜2 4 ˜m Z ∗ Z + J∗(Z − iaπ) + (Z∗+ iaπ∗) J # , (3.1.2) where ˜ B = 2B 2 + aeB, m =˜ 4m (2 + aeB)2, θ = θ + 2a.˜ (3.1.3)

If we compare this with (2.2.9), we see that this is just another non-commutative theory with ˜θ = θ+2a. The only difference from the standard non-commutative generating functional is the coupling between π and the source J.

If we introduce a further change of variables

π = ˜π + ia∆−1J, (3.1.4) where ∆ = 1 ˜ m + i˜θ∂t , (3.1.5)

then we have standard renormalized source terms with an additional quadratic source term, provided we set

a = 1

e ˜B − θ

2. (3.1.6)

The action is then S = Z ∞ −∞ dt " ˜ π∗ Z −˙ ie ˜B 2 ˜mZ ! + Z˙∗₊ ie ˜B 2 ˜mZ ∗ ! ˜ π −˜π∗ 1 ˜ m + i˜θ∂t ˜ π −e 2_B_˜2 4 ˜m Z ∗ Z + 1 −a ˜ θ J∗Z + 1 − a ˜ θ Z∗J + a2J∗∆−1J . (3.1.7) If we solve equations (3.1.3) and (3.1.6) we find

˜ θ = 4 − θeB eB ˜ B = 2B 4 − θeB (3.1.8) ˜ m = 4m (4 − θeB)2 a = 2 − θeB eB . (3.1.9)

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CHAPTER 3. THE EXACT RENORMALIZATION GROUP (ERG) AND

NON-COMMUTATIVE DUALITIES 25

We now have the exact relation between the generating functionals of two non-commutative theories Zθ[J ] = eia 2R∞ −∞dt[J ∗_∆−1_J ] ˜_Z_˜ θ[κJ ] (3.1.10) with κ = 2 4 − θeB. (3.1.11)

In the special case of θ = 0, this gives the commutative/non-commutative duality Zc[J ] = eia 2R∞ −∞dt[J ∗_∆−1_J ]Znc 1 2J , (3.1.12)

where the non-commutative theory has the non-commutative parameter ˜θ =

4

eB. Now we have a complete duality between the commutative and

non-commutative theories. No zero mass limit or infinite magnetic field limit had to be taken as is normally done. This duality is exact.

From (3.1.12) the following simple rule tells how to compute the correlators of the non-commutative theory in terms of the correlators of the commutative theory, and vice versa,

δ δJ → δ δJ + ia 2 (∆∗)−1J∗, δ δJ∗ → δ δJ∗ + ia 2 ∆−1J. (3.1.13)

since these equations can be easily inverted. We now have a complete dictio-nary for the computation of correlation functions.

3.2 The Exact Renormalization Group (ERG)

The duality above can be constructed in a more general way [41], and is dis-cussed in detail for the Landau problem in [41], where the ERG is used to construct dual families of non-commutative theories. We now review this con-struction which we will use in the next chapter for the complex scalar field.

The basic idea of the ERG procedure is to modify the kinetic energy in a particular way [43] to avoid ultraviolet divergences by introducing a cut-off. This modification is defined by a parameter called the flow parameter. In the ERG construction the interacting part of the action is also modified in such a way that the generating functional doesn’t depend on the flow parameter, at least below the cut-off scale. The change of the interaction part of the action is defined by a set of flow equations. In this way we have not changed the physics below the cut-off. We also have to restrict the sources so they are zero above the cut-off. When the flow parameter is zero we obtain the standard action which defines the initial conditions of the flow equations. A detailed discussion

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can be found in the textbook [43]. We can use the ERG construction to build our non-commutative/commutative duality.

Essentially the commutative and non-commutative theories differ in the kinetic energy terms, this is clear when comparing (1.1.28) with (2.2.8). We can use the ERG to derive a duality between these theories, if we identify the flow parameter as the non-commutative parameter θ. The only difference from the standard ERG approach is that the restricting assumptions on the modified kinetic energy term and source terms do not necessarily apply in this case [43]. We can compensate for this by introducing flow equations for the source terms, derived in [41] and summarized in appendix D.

We consider a complex scalar field theory in 0 + 1-dimensions with Fourier transformed action

S[φ, φ∗] = Z

dω φ∗(ω)K(ω, `)φ(ω) + SI[φ, φ∗] + J`[φ, φ∗]. (3.2.1)

Here K(ω, `) is our modified kinetic term and ` the flow parameter. The kinetic energy takes the standard form

K(ω, 0) = mω2, (3.2.2)

when the flow is switched off. The second term SI[φ, φ∗]is the interacting part

of the action, it is a function of the flow parameter `. J`[φ, φ∗]is a generalized

source term, which is a functional of the fields, determined by the requirement of invariance of the generating functional. We will see, when we flow the source, that it is sufficient to take this source term to be linear if the interaction is at most quadratic in the fields

J`[φ, φ∗] =

Z

dω [J0(l) + J0∗(l) + J1(l)φ∗(ω) + J1∗(l)φ(ω)]. (3.2.3)

The initial conditions imply J0(0) = J0∗(0) = 0 and J1(0) is an arbitrary

function of ω that acts as the standard source in the bare (` = 0) action. We now apply the logic of the ERG as set out in [44] and [43], and require Z[J ] = W [J ]_{W [0]} to be invariant under the flow, i.e. independent of `. However, we relax the usual conditions imposed on K(ω, `) and the sources, that lead to the weaker form of duality mentioned earlier, which necessitates the flow of the source terms as well. A derivation of the generalized flow equations can be found in appendix D. The flow equations for the interacting part and source terms are then given by

∂`SI = Z dω ∂`K−1 δSI δφ∗_(ω) δSI δφ(ω) − δ2_S I δφ∗_(ω)δφ(ω) (3.2.4)

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CHAPTER 3. THE EXACT RENORMALIZATION GROUP (ERG) AND NON-COMMUTATIVE DUALITIES 27 ∂`J` = Z dω ∂`K−1 δSI δφ(ω) δJ` δφ∗_(ω) + δSI δφ∗_(ω) δJ` δφ(ω) + δJ` δφ∗_(ω) δJ` δφ(ω) − δ2_J ` δφ∗_(ω)δφ(ω) . (3.2.5)

These equations can easily be solved when the interaction term is quadratic in the fields, i.e.

SI[φ, φ∗] =

Z

dω φ∗(ω)g(ω, `)φ(ω), (3.2.6)

with g(ω, `) real. Focusing on the source terms for the moment and using (3.2.5) and (3.2.3)

∂`J1(`) = ∂`K−1(ω, `)g(ω, `)J1(`),

∂`[J0(`) + J0∗(`)] = ∂`K−1(ω, `)|J1(`)|2. (3.2.7)

Integrating these equations and using the initial conditions on the sources, we obtain J1(`) = J1(0) exp Z ` 0 d`0 g(ω, `0)∂`0K−1(ω, `0) , J0(`) + J0∗(`) = |J1(0)|2 Z ` 0 d`0 ∂`0K−1(ω, `0) × exp 2 Z `0 0 d`00 g(ω, `00)∂`00K−1(ω, `00) ! . (3.2.8)

We now apply this result to the Landau problem (1.1.28), for which the Fourier transformed action reads

S =

Z

dω [z∗(ω)mω2z(ω) − eBωz∗(ω)z(ω)

+J (ω)z∗(ω) + J∗(ω)z(ω)]. (3.2.9)

Motivated by the form of the action for the non-commutative Landau problem (2.2.7) we take

K(ω, `) = mω

2

(1 − mω` − i), (3.2.10)

the epsilon term ensures the convergence of the path integral, but hereafter we drop it. The interacting part of the action (3.2.6) has the initial condition

g(ω, 0) = −eBω. (3.2.11)

Inserting (3.2.9) into (3.2.4) we obtain the following equation for the interact-ing term ∂g(ω, `) ∂` = − 1 ωg 2 (ω, `). (3.2.12)

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Note we have ignored a vacuum term generated from the right hand side of (3.2.4), since this just adds a term independent of z to the interaction. Solving the equation with the initial conditions imposed produces

g(ω, `) = e ˜B(`)ω, B(`) =˜ B

1 + eB`. (3.2.13)

From (3.2.5) we obtain for the source, J1(`) = J1(0) 1 − eB`, J0(`) + J0∗(`) = − |J1(0)|2` ω(1 − eB`). (3.2.14)

However, this action still does not quite have the same form as the non-commutative action so we rescale the fields in an ω-independent way

˜ z(ω) =

r 1

1 − eB`z(ω). (3.2.15)

This yields the dual action [41]

S = Z dω ˜ z∗(ω)K(ω, `)˜z(ω) − eBω (1 − mω`)z˜ ∗ (ω)˜z(ω) − |J1(0)| 2_` ω(1 − eB`)+ J1(0) √ 1 − eB`z˜ ∗ (ω) + c.c. . (3.2.16)

Now the generating functional is seemly not independent of `, however this is just a change of variables in the path integral, if done correctly this does not change the propagator. Care must be taken when computing correlators, as one needs to keep in mind which fields we are calculating the correlators of. Since we have a dictionary between the commutative and non-commutative fields this poses no problem. Now this is the action of a non-commutative Landau problem with

θ = ` (3.2.17)

and a magnetic field ˜B determined by B = ˜B 1 − e ˜B`

4 !

. (3.2.18)

We have a duality similar in form to (3.1.12), and thus (3.2.16) represents a family of dual non-commutative theories. By duality we mean that for every commutative system with magnetic field B there is a corresponding family of non-commutative systems with magnetic field ˜B.

Using these values (3.2.18) of the non-commutative parameter and mag-netic field in the excitation energy (2.2.12) indeed yields the commutative

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CHAPTER 3. THE EXACT RENORMALIZATION GROUP (ERG) AND

NON-COMMUTATIVE DUALITIES 29

cyclotron frequency ωc= eB_m as is required by the duality. We have explicitly

computed the generating functionals for the non-commutative and commuta-tive theories, which is simple to do since we have a Gaussian path integral. We do indeed find the generating functions are identical. It should also be noted that the flow equations are not unique, as explained in [45]. In this paper the relationship between coarse graining and the ERG is discussed. In fact, there is a map between different kinds of coarse graining and different flow equations. We discuss this in the next chapter when we investigate the fate of symmetries in the ERG procedure.

3.3 Chapter Summary

In this chapter we reviewed the ERG construction for the quantum mechanical Landau problem, as a field theory in 0 + 1 dimensions. Also in this chapter we have constructed a more general duality between non-commutative theories using a change of variables. This was interesting in its own right and also allowed us to benchmark our ERG construction. Now we are well prepared to tackle the ERG construction for field theories and also interacting theories. This is the subject of the next chapter.

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Chapter 4 Non-Canonical ERG Dualities

We can now proceed to the ultimate goal of this part of the thesis which is to develop dualities for non-canonical field theories. This work appears in the publication [46]. We start with the free complex scalar theory that proceeds in a very similar way to the 0+1 dimensional field theory of quantum mechanics in the previous chapter. We also study the fate of the symmetries in this duality using a more general flow [45]. This is important as tracking symmetries, at least in principle, is one of the motivations for these dualities. Finally we study the most complicated duality covered in this thesis for the interacting complex scalar field.

4.1 Free Complex Scalar Field Theory

We now follow the ERG procedure as described in section 2.2.1. We again modify the kinetic energy term, based on the action of non-canonical field the-ories. We then let the interaction flow, thus leaving the normalized generating functional unchanged. In four dimensions we can write down the action that results from the ERG procedure as

S[φ, φ∗] = Z

d4k φ∗(k0, ~k)K(k0, θ)φ(k0, ~k) + SI[φ, φ∗] + Jθ[φ, φ∗], (4.1.1)

we have also preemptively identified the flow parameter with the non-commutative length scale θ as in (3.2.17). For the non-canonical complex scalar field theo-ries [21; 22] in four dimensional Minkowski space, the action in Fourier space is given by S = Z d4k φ∗(k0, ~k) (k 0₎2 1 − θk0 −~k 2 + m2 ! φ(k0, ~k), (4.1.2) which amounts to the modification

(k0)2 → K(k0_{, θ) =} (k0)2

1 − θk0. (4.1.3)

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CHAPTER 4. NON-CANONICAL ERG DUALITIES 31

Some discussion of the critical point k0 ₌ 1

θ can found in [21]. Solving the

ERG as in [41], we have the flow equations, (3.2.4) and (3.2.5), adapted to four dimension, these are derived in appendix D

∂θSI = Z d4k ∂θK−1 δSI δφ∗_(k) δSI δφ(k) − δ2SI δφ∗_(k)δφ(k) , (4.1.4) ∂θJθ = Z d4k ∂`K−1 δSI δφ(k) δJ δφ∗_(k) + δSI δφ∗_(k) δJ δφ(k) + δJ δφ∗_(k) δJ δφ(k) − δ2_J δφ∗_(k)δφ(k) . (4.1.5)

Using the flow equations the remaining quadratic term flows as g(θ) =

k0~k2_{+ m}2

θ(~k2_{+ m}2_{) − k}0. (4.1.6)

For the free theory we have

SI[φ, φ∗] = φ∗(k0, ~k)g(θ)φ(k0, ~k) (4.1.7)

which has the initial condition

g(0) = −~k2+ m2. (4.1.8)

Note we have ignored a vacuum term generated from the right hand side of (4.1.4), since this just adds a term independent of φ and J to the interaction. The source term becomes

J1(0) → J1(θ) =

k0J1(0)

k0_{− θ(~k}2_{+ m}2₎, (4.1.9)

with an additional quadratic term J0(θ) = −

θ|J1(0)|2

k0_{− θ(~k}2_{+ m}2₎. (4.1.10)

The full action with source terms is therefore Sθ =

Z

d4k [φ∗Kφ + φ∗gφ + J [φ, φ∗]] . (4.1.11) This action is not quite that of the non-canonical system, this same effect occurred in the quantum mechanical case. Again it can be brought into the

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right form if we rescale the fields. However, this time it is a momentum dependent rescaling of the fields

φ(k0, ~k) → s

K(θ) + g(0) K(θ) + g(θ)φ(k

0_{, ~k).} _(4.1.12)

This of course also modifies the sources. Now we have again a complete duality between the commutative and non-canonical field theories with generating functionals independent of the non-commutative parameter θ, just like section 2.2.1.

As we mentioned earlier, and as explained in [24] and [45] we can think of the ERG as a coarse graining or blocking procedure i.e a transformation of the fields. To see this, recall that it is possible to interpret the renormalization flow as a change of variables in the path integral. If we ignore the source terms, then in our case the blocking procedure is a simple momentum dependent rescaling of the commutative fields

φ0(k0, ~k) =

s

K(θ) + g(θ) K(0) + g(0)φθ(k

0_{, ~k).} _(4.1.13)

This immediately produces the non-commutative action with modified sources without having to solve the flow equations. However, in position space the momentum dependent rescaling above is highly non-local. In the interacting case, this blocking procedure becomes a much more complicated transforma-tion, not even necessarily a linear one, and so in this case the flow equations become a necessity.

4.2 Symmetries

Before moving on to the interacting case, we investigate the fate of the Lorentz symmetry under the dualities constructed via the ERG and the corresponding coarse graining. We investigate the simplest case of a momentum dependent rescaling of the fields, such as in (4.1.13).

The two theories are dual, as is ensured by the ERG flow equations, and therefore the symmetry must be present in both. It may, however, not be manifest in the dual theory but instead be implemented in a highly non-trivial way due to the coarse graining not being manifestly Lorentz covariant. In fact, it should already be clear from (4.1.12) that the Lorentz symmetry cannot be manifest in the dual theory and must be implemented in a highly non-local way.

A more general formulation of the flow equations which establishes the link to coarse graining [45] is − ∂θe−Sθ[φ]= Z d4k δ δφ(k) Ψθ[φ(k)]e −Sθ[φ] , (4.2.1)

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CHAPTER 4. NON-CANONICAL ERG DUALITIES 33

where Ψθ is a general functional of the fields. If the action satisfies this flow

equation, the normalized generating functional is again invariant, since the right-hand side of (4.2.1) leads to a total derivative of the integrand of the path integral. If we relax the conditions on the kinetic term and the sources, we must also make sure the sources are adjusted appropriately.

Every allowed Ψθ is related to some blocking transformation (coarse

grain-ing) [45]. If we write the blocking transformation as fθ[φ0](k) = φ(k), where

φ0 and φ are the fields before and after the blocking respectively. Then we can

write an equation to calculate the corresponding Ψθ:

Ψθ[φ(k)]e−Sθ[φ]=

Z

[dφ0] δ [φ − fθ[φ0]] (∂θfθ[φ0]) e−S[φ0]. (4.2.2)

For the simplest coarse graining which is just multiplicative and invertible, for example (4.1.13), then

fθ[φ0](k) = fθ(k)φ0(k), (4.2.3)

implies

Ψθ[φ(k)] = |f_θ−1(k)|∂θfθ(k)φ(k). (4.2.4)

Then equation (4.2.1) gives a simple equation for the flow of the sources, if we include them in the action,

∂θJθ = |fθ−1(k)|∂θfθ(k)Jθ. (4.2.5)

The symmetries are implemented differently in the two theories. If the fields transform as φ0 → ˜φ0 under a symmetry transformation in the original

theory, the transformation induced in the dual theory is

φ → ˜φ = fθ(k) ˜φ0(k). (4.2.6)

Applying this to the Lorentz generators, we find them to be given in the dual theory by Mij = ki∂˜kj− kj∂˜_ki (4.2.7) where ˜ ∂ki = ∂_ki− ∂kif (k) f (k) . (4.2.8)

Note that in real space this corresponds to a highly non-local implementation of the Lorentz symmetry. As expected, the symmetry exists in the dual theory, however, implemented in a different way. When we have a more complicated coarse graining, such as the interacting case, it may prove impractical to track the symmetry in this way.

Lorentz symmetry in non-commutative field theories : commutative/non-commutative dualities and manifest covariance

Manifest Covariance

by

Paul Henry Williams

Dissertation presented for the degree of Doctor of Philosophy

in Physics in the Faculty of Natural Science at Stellenbosch

University

Declaration

Abstract

Lorentz

Symmetry in Non-Commutative Field Theories:

Commutative/Non-Commutative

Dualities and Manifest

Covariance

Uittreksel

Lorentz

Simmetrie in Nie-kommutatiewe Veldeteorieë:

Kommutatiewe/Nie-kommutatiewe

Dualiteite en

Eksplisiete

Kovariansie

Acknowledgements

Contents

I Non-commutative/Commutative Dualities

12

II SU(1,1) Field Theory

38

Chapter 1

Introduction

1.1

Commutative Quantum Mechanics

1.1.1

The Schrödinger Equation

1.1.2

The Path Integral

1.2

Commutative Field Theory

Part I

Non-commutative/Commutative

Dualities

Chapter 2

Non-Commutative and

Non-Canonical Theories

2.1

Non-Commutative Quantum Mechanics

2.2

Position and the Path Integral

2.2.1

The Landau Problem

2.3

Non-Canonical Field Theory

2.3.1

Complex Field Theory

2.4

Chapter Summary

Chapter 3

The Exact Renormalization Group

(ERG) and Non-Commutative

Dualities

3.1

Bopp Shift

3.2

The Exact Renormalization Group (ERG)

3.3

Chapter Summary

Chapter 4

Non-Canonical ERG Dualities

4.1

Free Complex Scalar Field Theory

4.2

Symmetries